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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 802585 6 pageshttpdxdoiorg1011552013802585
Research ArticleInverse Problem Models of Oil-Water Two-Phase Flow Based onBuckley-Leverett Theory
Rui Huang1 Xiaodong Wu1 Ruihe Wang2 and Hui Li3
1 College of Petroleum Engineering China University of Petroleum Beijing 102249 China2 China National Oil and Gas Exploration and Development Corporation Beijing 100034 China3 CNPC Beijing Richfit Information Technology Co Ltd Beijing 100013 China
Correspondence should be addressed to Rui Huang huangrui25126com
Received 5 September 2013 Revised 30 October 2013 Accepted 30 October 2013
Academic Editor Xiaoqiao He
Copyright copy 2013 Rui Huang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Based on Buckley-Leverett theory one inverse problemmodel of the oil-water relative permeability was modeled and proved whenthe oil-water relative permeability equations obey the exponential form expression and under the condition of the formationpermeability that natural logarithmdistribution always obey normal distribution the other inverse problemmodel on the formationpermeability was proved These inverse problem models have been assumed in up-scaling cases to achieve the equations byminimization of objective function different between calculation water cut and real water cut which can provide a reference forresearching oil-water two-phase flow theory and reservoir numerical simulation technology
1 Introduction
Reservoir numerical simulation technology is a growing newdiscipline with the emergence and development of the com-puter technology and computational mathematics which hasachieved rapid development and wide application all over theworld for example the study of reservoir numerical simu-lation based on formation parameters [1] the well modelsand impacts [2] a numerical simulator for low-permeabilityreservoirs [3] and so on However the majority of methodshave a great amount of calculation It will waste a lotof time and energy when we research the oil-water relativepermeability equations or the formation permeability distri-butionTherefore according to the inverse problemmodelingbased on oil-water two-phase flow we propose a method toobtain the relevant results for our research It is importantto define the formation permeability distribution and oil-water relative permeability equations which provide neces-sary information for oil reservoir evaluation for examplelarge-scaling evaluation which could be applied in reservoirnumerical simulation during evaluation
In order to get the answers which will be able to apply inlarge-scaling cases and solve the corresponding problems in
reservoir scaling inverse problemmodels have to be assumedto achieve the equations by minimization of objective func-tion differently between real water cut and calculation watercut The theoretical grid model is shown in Figure 1
If we know the distribution 1198701 1198702 119870
119899 then the
inverse problem model will be as shown in (1)The first question The optimization distribution
119886119908 119887119908 119886119900 119887119900 and the relative permeability equations
119870119903119908(119878119908)119870119903119900(119878119908)
The inverse problem mathematical model on oil-waterrelative permeability is as followsobjective function
119864 = min119899119905
sum119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2
where 119899119905 is time steps
initial condition
119878119908
119894(0)
119895= 119878119868119908119895
1198760
119895= 0
0 le 119870119903119900119894
119895lt 1 0 le 119870119903119908
119894
119895lt 1
2 Journal of Applied Mathematics
1234
n
234m
The average partition total m
The p
artit
ion
tota
l n
Supply edge Fluid producing edge
middot middot middot
re
Figure 1 Reservoir profile model
119870119903119908
(119878119908) = 119886119908(
119878119908minus 119878119908119888
1 minus 119878119908119888
minus 119878119900119903
)
119887119908
119870119903119900(119878119908) = 119886119900(1 minus 119878119900119903minus 119878119908
1 minus 119878119908119888
minus 119878119900119903
)
119887119900
if 119903119890gt 119903119908119891
119894
119895 then 119878
119908
119894(119905)
119895= 119878119868119908119895
if 119903119890le 119903119908119891
119894
119895 then 119878
119908
119894(119905)
119895= 119891minus1
119908
1015840
(119878119908)
119891119908= 119891119908(119878119908)
mathematical model
119870119903119900119894
119895= 119870119903119900 (119878
119908
119894(119905)
119895) 119870119903119908
119894
119895= 119870119903119908 (119878
119908
119894(119905)
119895)
119879119894
119895=
2120587ℎ119895119870119895
ln (119903119903119894119895)(119870119903119900
119894
119895
120583119900
+119870119903119908
119894
119895
120583119908
)
119879119895=
1
sum119898
119894=1(1119879119894119895) 119879 =
119899
sum119895=1
119879119895 119876
119905
119895=119879119895
119879sdot 119876119905
1199032
119890minus 119903119894
119895
2
=119891119905
119908(119878119908
119894
119895)1015840
119876119905
119895
120601119895sdot 120587ℎ119895
119903119908119891
2
119895= 1199032
119890minus119891119905
119908(119878119908119891
119894
119895)1015840
119876119905
119895
120601119895sdot 120587ℎ119895
119876119905
119908119895= 119876119905
119895sdot 119891119905
119908(1198781
119908119895) 119876
119905
119900119895= 119876119905
119895sdot (1 minus 119891
119905
119908(1198781
119908119895))
119876119905
119908=
119899
sum119895=1
119876119905
119908119895 119876
119905
119900=
119899
sum119895=1
119876119905
119900119895 119891
119908(119905) =
119876119905
119908
119876119905119908+ 119876119905119900
(1)
If we know the equations 119870119903119908(119878119908) 119870119903119900(119878119908) another
inverse problem model will be obtained as shown in (2)The second question an optimization distribution prob-
lem 1198701 1198702 119870
119899 and the standard deviation 120590
The inverse problem mathematical model of the forma-tion permeability is as follows
objective function
119864 = min119899119905
sum119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2
where 119899119905 is time steps
initial condition
119878119908
119894(0)
119895= 119878119908119888119895
1198760
119895= 0
119891 (119870119895) =
1
radic2120587 sdot 120590119890minus(ln119870
119895minus120583)2
21205902
120583 = Ln119870
if 119903119890gt 119903119908119891
119894
119895 then 119878
119908
119894(119905)
119895= 119878119908119888119895
if 119903119890le 119903119908119891
119894
119895 then 119878
119908
119894(119905)
119895= 119891minus1
119908
1015840
(119878119908)
119891119908= 119891119908(119878119908)
mathematical model
119870119903119900119894
119895= 119870119903119900 (119878
119908
119894(119905)
119895) 119870119903119908
119894
119895= 119870119903119908 (119878
119908
119894(119905)
119895)
119879119894
119895=
2120587ℎ119895119870119895
ln (119903119903119894119895)(119870119903119900
119894
119895
120583119900
+119870119903119908
119894
119895
120583119908
)
119879119895=
1
sum119898
119894=1(1119879119894119895) 119879 =
119899
sum119895=1
119879119895 119876
119905
119895=119879119895
119879sdot 119876119905
1199032
119890minus 119903119894
119895
2
=119891119905
119908(119878119908
119894
119895)1015840
119876119905
119895
120601119895sdot 120587ℎ119895
119903119908119891
2
119895= 1199032
119890minus119891119905
119908(119878119908119891
119894
119895)1015840
119876119905
119895
120601119895sdot 120587ℎ119895
119876119905
119908119895= 119876119905
119895sdot 119891119905
119908(1198781
119908119895) 119876
119905
119900119895= 119876119905
119895sdot (1 minus 119891
119905
119908(1198781
119908119895))
119876119905
119908=
119899
sum119895=1
119876119905
119908119895 119876
119905
119900=
119899
sum119895=1
119876119905
119900119895 119891
119908(119905) =
119876119905
119908
119876119905119908+ 119876119905119900
(2)
Here 119891(history)119908
(119905) is history water cut and 119891119908(119905) is cal-
culation water cut the first water cut is a known amountfrom production and another is calculated by our inverseproblem models Water cut means water production rate is akey parameter in reservoir engineering which can representwater production capacity in reservoir development If watercut is too high it may bring negative effects to reservoirproduction Therefore the history matching for water pro-duction rate plays an important role in reservoir dynamicanalysis and numerical simulation [4ndash6]
And thus 119894 = 1 2 3 119898 119895 = 1 2 3 119899 119898
is the average partition total 119899 is the partition total 119879is grid conductivity ℎ
119895is each longitudinal layer stratum
Journal of Applied Mathematics 3
dr
Supp
ly ed
ge
Fluid producing edge
h
re
Figure 2 Radial flow unit model
thickness 119870119895is each longitudinal layer permeability 120601
119895is
each longitudinal layer porosity 119878119894119908119895
is grid water saturationat different time 119878
119908119888is the initial water saturation 119864 is
the objective function 119903119890is reservoir radius 119870 is average
permeability 119870119903119900
is oil relative permeability 119870119903119908
is waterrelative permeability119876
119905is the total fluid production119876119905
119908is the
total water yield of fluid producing edge at different time 119876119905119900
is the total oil production of fluid producing edge at differenttime 119876119905
119895is the total fluid production of each longitudinal
layer at different time 119878119908119891
is the frontier saturation 119903119908119891
is thecorresponding displacement of frontier saturation
The inverse models (1) and (2) contain Buckley-Leveretttheory of two-phase flow [7] and establish the oil-water two-phase plane radial flow function 1199032
119890minus 1199032= 1198911015840
119908sdot 119876119905120601 sdot 120587 sdot ℎ
without considering these factors of gravity and capillarypressure in addition that shows the movement rules ofisosaturation surface As said above most researchers alwaysrely on forward problems [8] of core sampling experimentand mathematical statistics for an answer As a result thereis always a large error between the calculation water cut andthe history water cut in reservoir water cut analysis In thispaper according to historical water cut and calculation watercut to establish the objective function 119864 = minsum119899119905
119905=1(119891119908(119905) minus
119891(history)119908
(119905))2 By the end an inverse problem model on the
optimal solution distribution 119886119908 119887119908 119886119900 119887119900 and oil-water
relative permeability equations 119870119903119908(119878119908) 119870119903119900(119878119908) based on
the Buckley-Leveret theory of two-phase flow can be realizedand another inverse problem model was modeled under thecondition of the formation permeability logarithmic functionare always obey normal distribution Finally it can provide akey information for reservoir numerical simulation studies
2 Mathematical Model of Oil-WaterTwo-Phase Plane Radial Flow
Theorem 1 Without considering these factors of gravity andcapillary pressure through the Buckley-Leveret theory of two-phase flow one established the oil-water two-phase plane radialflow mathematical model
1199032
119890minus 1199032=1198911015840
119908sdot 119876119905
120601 sdot 120587ℎ (3)
Proof We suppose the liquid flow rule is a plane radial flowfrom reservoir limit to well and choose a volume element inthe vertical direction of streamline as shown in Figure 2
According to the seepage principle [9] we can get a flowequation of the volume element
119902119908= 119902 sdot 119891
119908 (4)
In the 119889119905 time the flow volume of the volume element is
119876out = 119889119902119908sdot 119889119905 (5)
We can derive from (4) and (5) that
119876out = 119902 sdot 119889119891119908sdot 119889119905 (6)
Meanwhile the inflow volume of the volume element is
119876in = 120601 sdot 2120587ℎ sdot 119903119889119903 sdot 119889119904119908 (7)
In the 119889119905 time relying on the Buckley-Leveret theory oftwo-phase flow 119876in = 119876out from (6) and (7) we have that119902 sdot 119889119891119908sdot 119889119905 = 120601 sdot 2120587ℎ sdot 119903119889119903 sdot 119889119878119908 as follows
119902 sdot119889119891119908
119889119878119908sdot 119889119905 = 120601 sdot 2120587ℎ sdot 119903119889119903 (8)
From 119891119908= 119891119908(119878119908) we can get that 1198911015840
119908= 119889119891119908119889119878119908
Then deriving from (8) that 119902 sdot119889119891119908(119878119908) sdot119889119905 = 120601 sdot2120587ℎ sdot119903119889119903
after infinitesimal calculus we obtain
1198911015840
119908int119905
0
119902 sdot 119889119905 = 120601 sdot 2120587ℎ sdot int119903119890
119903
119903119889119903 (9)
If119876119905 = int119905
0119902 sdot119889119905 then (9) turns to1198911015840
119908sdot119876119905= 120601sdot120587ℎ sdot (119903
2
119890minus1199032)
We obtain
1199032
119890minus 1199032=1198911015840
119908sdot 119876119905
120601 sdot 120587ℎ (10)
From the reservoir profile grid model the plane radial flowmodel of the grids in the longitudinal each layer can showsthat
1199032
119890minus 119903119894
119895
2
=119891119905
119908(119878119908
119894
119895)1015840
119876119905
119895
120601119895sdot 120587ℎ119895
(11)
Theorem 2 A special saturation definition about ldquo119878119908119891rdquo from
the water cut function curve 119891119908
= 119891119908(119878119908) selecting the
point of the irreducible water saturation as a fixed point andjoining any other point on the curve and constructing function119896(119878119908119891) = (119891
119908(119878119908119891)minus119891119908(1198781199081))(119878119908119891minus1198781199081) ifMax119896(119878
119908119891) then
one can call ldquo119878119908119891rdquo frontier saturation Now the corresponding
displacement of frontier saturation 119903119908119891
is
119903119908119891
2
119895= 1199032
119890minus119891119905
119908(119878119908119891
119894
119895)1015840
119876119905
119895
120601119895sdot 120587ℎ119895
(12)
4 Journal of Applied Mathematics
3 The Inverse Problem Model on Oil-WaterRelative Permeability
31 The Oil-Water Relative Permeability Equations In thereservoir engineering the different lithological character hasthe different corresponding oil-water relative permeabilitycurve equations [10] but the most commonly form is usedas a kind of the exponential form expression as follows
119870119903119908
(119878119908) = 119886119908(
119878119908minus 119878119908119888
1 minus 119878119908119888
minus 119878119900119903
)
119887119908
119870119903119900(119878119908) = 119886119900(1 minus 119878119900119903minus 119878119908
1 minus 119878119908119888
minus 119878119900119903
)
119887119900
(13)
In reservoir profile grid model the oil-water relative perme-ability curve equations of each grid can be showed
119870119903119908
(119878119894
119908119895) = 119886119908(
119878119894
119908119895minus 119878119908119888
1 minus 119878119908119888
minus 119878119900119903
)
119887119908
119870119903119900(119878119894
119908119895) = 119886119900(1 minus 119878119900119903minus 119878119894
119908119895
1 minus 119878119908119888
minus 119878119900119903
)
119887119900
(14)
In the equations we can know different 119886119908 119887119908 119886119900 119887119900 corre-
sponding to its own 119870119903119908(119878119908) 119870119903119900(119878119908)
32 The Inverse Problem Mathematical Model of Oil-WaterRelative Permeability Equations
Theorem 3 If the oil-water relative permeability equations119870119903119908(119878119908) 119870119903119900(119878119908) always obey (13) and one can calculate the
water cut 119891119908(119905) based on the model of the Theorem 1 if the
objective function 119864 = minsum119899119905119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2 can
be satisfied then the inverse problem model (1) will have theoptimal solution of the distribution 119886
119908 119887119908 119886119900 119887119900 and oil-
water relative permeability equations 119870119903119908(119878119908) 119870119903119900(119878119908)
Proof Relying on oil-water relative permeability equations(14) and the conductivity definition of numerical reservoirsimulation [11] we get the following
initial value 1198860
119908 1198870
119908 1198860
119900 1198870
119900
119870119903119908
(119878119908) = 1198860
119908(
119878119908minus 119878119908119888
1 minus 119878119908119888
minus 119878119900119903
)
1198870
119908
119870119903119900(119878119908) = 1198860
119900(1 minus 119878119900119903minus 119878119908
1 minus 119878119908119888
minus 119878119900119903
)
1198870
119900
119879119894
119895=
2120587ℎ119895119870119895
ln (119903119903119894119895)(119870119903119900
119894
119895
120583119900
+119870119903119908
119894
119895
120583119908
)
119879119895=
1
sum119898
119894=1(1119879119894119895) 119879 =
119899
sum119895=1
119879119895
119876119905
119895=119879119895
119879sdot 119876119905
(15)
We can derive from (12) and (15) 119903119908119891
If 119903119890gt 119903119908119891 then 119878
119908
119894(119905)
119895= 119878119908119888119895
If 119903119890le 119903119908119891
solved inversefunction of (11) and calculated water saturation of the fluidproducing edge then 119878
119908
119894(119905)
119895= 119891minus1
119908
1015840
(119878119908) and so forth 119894 = 1
If we obtain the value of the water saturation [12] relyingon the function 119891
119908(119878119908) sim 119878
119908 we can calculate the value
of the longitudinal each layer water cut 119891119905119908(1198781
119908119895) and the
constructing mathematic model as follows
119876119905
119908119895= 119876119905
119895sdot 119891119905
119908(1198781
119908119895)
119876119905
119900119895= 119876119905
119895sdot (1 minus 119891
119905
119908(1198781
119908119895))
119876119905
119908=
119899
sum119895=1
119876119905
119908119895 119876
119905
119900=
119899
sum119895=1
119876119905
119900119895
119891119908(119905) =
119876119905
119908
119876119905119908+ 119876119905119900
(16)
From (16) we can calculate the total water cut of the fluidproducing edge 119891
119908(119905) and rely on the objective function 119864 =
minsum119899119905119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2 with numerical optimization
calculation in the inverse problem model an optimizationproblem of the distribution 119886
119908 119887119908 119886119900 119887119900 will be obtained
so the 119870119903119908(119878119908) and 119870
119903119900(119878119908) can be formed The proof of
Theorem 3 is completed
4 The Inverse Problem Model on theFormation Permeability LogarithmicFunction Always Obey Normal Distribution
Theorem 4 If the formation permeability logarithmic func-tion distribution Ln119870
1 Ln119870
2 Ln119870
119899 always obey nor-
mal distribution 119891(119870) = (1radic2120587 sdot 120590)119890minus(ln119870minus120583)221205902 120583 =
Ln119870 meanwhile one calculates the value of the 119891119908(119905) based
on the model of Theorem 1 which can be adapted to theobjective function 119864 = minsum119899119905
119905=1(119891119908(119905)minus119891
(history)119908
(119905))2 then the
inverse problem model (2) will have the optimal distributionLn119870
1 Ln119870
2 Ln119870
119899 and 119870
1 1198702 119870
119899
Proof According to 3120590 principle for normal distributioncurve if 119875(120583minus3120590minus119886 lt 119883 le 120583+3120590+119886) rarr 1 then there are aleft end point value and a right end point value on the curvethe119883 value of the left end point is ln119870min or 120583 minus 3120590 minus 119886 The119883 value of the right end point is ln119870max or 120583 + 3120590 + 119886 Andthus 119886 gt 0 and 120583 = ln119870 = 05 sdot (ln119870min + ln119870max)
According to the area superposition principle of normaldistribution curve Select 119909 = ln119870 as the starting point stepsize Δ119909 and make a subdivision for probability curve if thearea summation of these formed closed figures can infinitelyapproach 1 then we can obtain the ln119870min value of the leftend point and the ln119870max value of the right end point andln119870max = 2 sdot ln119870 minus ln119870min
According to the area superposition principle of normaldistribution curve and giving a initial value 120590 of normaldistribution then119883
0= ln119870min and119883119899 = ln119870max 119865(119883119895) is a
cumulative distribution function of the normal distribution[13] If we use equal step size Δ119883 make a subdivision for
Journal of Applied Mathematics 5
probability curve then the area summation of every closedfigures 119878
119895= 119865(119883
119895) minus119865(119883
119895minus1) each longitudinal layer stratum
thickness ℎ119895= ℎ sdot 119878
119895 and 119883
119895= Δ119883 sdot 119895 + 119883
0 119870119895= 119890119883119895
if we use equal area Δ119878 make a subdivision for probabilitycurve then the area summation of every closed figures 119878
119895=
119865(119883119895)minus119865(119883
119895minus1) 119878119895= Δ119878 = 1119899 ℎ
119895= ℎsdot119878
119895 and119883
119895= 119865minus1(119878119895)
119870119895= 119890119883119895 And thus 119895 = 1 2 3 119899 (119899 is the total number of
vertical stratification)Relying on the distribution 119870
1 1198702 119870
119899 oil-water
relative permeability functions 119896119903119900 = 119896119903119900(119878119908) 119896119903119908 =
119896119903119908(119878119908) [14] the conductivity definition of numerical reser-
voir simulation we get the following equations
119896119903119900119894
119895= 119896119903119900 (119878
119908
119894(119905)
119895) 119896119903119908
119894
119895= 119896119903119908 (119878
119908
119894(119905)
119895)
119879119894
119895=
2120587ℎ119895119870119895
ln (119903119903119894119895)(119870119903119900
119894
119895
120583119900
+119870119903119908
119894
119895
120583119908
)
119879119895=
1
sum119898
119894=1(1119879119894119895) 119879 =
119899
sum119895=1
119879119895
119876119905
119895=119879119895
119879sdot 119876119905
(17)
We can derive from (12) and (17) that 119903119908119891119895
If 119903119890
le 119903119908119891119895
solving inverse function of (11) andcalculating water saturation of the fluid producing edge then119878119908
119894(119905)
119895= 119865minus1(119903119894
119895 120601119895 ℎ119895 119876119905
119895) and thus 119894 = 1 If 119903
119890gt 119903119908119891119895
then
119878119908
119894(119905)
119895= 119878119908119888119895
and thus 119894 = 1If we obtain the value of the water saturation (119878
119908) relying
on the function119891119908(119878119908) sim 119878119908 we can calculate the value of the
longitudinal each layer water cut 119891119905119908(1198781
119908119895) then mathematic
model can be constructed as follows
119876119905
119908119895= 119876119905
119895sdot 119891119905
119908(1198781
119908119895) 119876
119905
119900119895= 119876119905
119895sdot (1 minus 119891
119905
119908(1198781
119908119895))
119876119905
119908=
119899
sum119895=1
119876119905
119908119895 119876
119905
119900=
119899
sum119895=1
119876119905
119900119895
119891119908(119905) =
119876119905
119908
(119876119905119908+ 119876119905119900)
(18)
Equation (18) can calculate the total water cut of the fluidproducing edge 119891
119908(119905) and relies on the objective function
119864 = minsum119899119905119905=1
(119891119908(119905)minus119891
(history)119908
(119905))2With numerical optimiza-
tion calculation in the inverse problem model the standarddeviation 120590 and an optimization problem of the distributionLn1198701 Ln1198702 Ln119870
119899 will be obtained so the distribution
1198701 1198702 119870
119899 will be given The proof of Theorem 4 is
completed
5 Discussions and Conclusions
The different distribution 119886119908 119887119908 119886119900 119887119900 corresponds to a
group of oil-water relative permeability equations 119870119903119908(119878119908)
and 119870119903119900(119878119908) Combining with the objective function
119864 = minsum119899119905119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2 through the optimization
solution method it will finally bring a set of the optimumdistribution 119886
119908 119887119908 119886119900 119887119900 and the oil-water relative
permeability equations119870119903119908(119878119908) and119870
119903119900(119878119908)
According to the above inverse problem mathematicalmodel based on the definition of the normal distributiondifferent (120583
119894 1205902
119894) corresponds to its own normal distribution
curve with the certain expectation 120583119894 If 120590
119894goes up the
volatility of its normal distribution will be stronger accordingto the definition of standard deviation which will finally leadto the large differential permeability distribution namelythe strong heterogeneity With the certain expectation 120583
119894
each different value of 120590119894corresponds to a group of original
values of (120583119894 1205902
119894) which will yield a group of values of
1198701 1198702 119870
119899minus1 119870119899 namely the value of permeability of
each single formation And it will also work out liquidproduction water production and integrated water cut of thewhole liquid outlet of each single formation Combining withthe objective function 119864 = minsum119899119905
119905=1(119891119908(119905) minus 119891
(history)119908
(119905))2
it will finally bring a set of optimum normal distributionsof 119883 sim (120583
119894 1205902
119894) (Etc 119883 = ln119870) through the optimization
solution method Analysis of the changes of water drivingplace oil production and water production can be donethrough the related mathematical model
Finally the idea of constructing inverse problem modelsaccording to the historical dynamic production data canbe realized which attaches great importance to formationheterogeneity observation of water flooding front positionand prediction of dynamic producing performance
Acknowledgments
The authors gratefully thank the support of Canada CMGFoundation and the referees for valuable suggestions
References
[1] S X Bing ldquoReservoir numerical simulation study based onformation parametersrsquo time-variabilityrdquo Procedia Engineeringvol 29 pp 2327ndash2331 2012
[2] F A Dumkwu A W Islam and E S Carlson ldquoReview ofwell models and assessment of their impacts on numericalreservoir simulation performancerdquo Journal of Petroleum Scienceand Engineering vol 82-83 pp 174ndash186 2012
[3] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011
[4] S Haj-shafiei S Ghosh and D Rousseau ldquoKinetic stability andrheology of wax-stabilized water-in-oil emulsions at differentwater cutsrdquo Journal of Colloid and Interface Science vol 410 pp11ndash20 2013
[5] S Talatori and T Barth ldquoRate of hydrate formation in crudeoilgaswater emulsions with different water cutsrdquo Journal ofPetroleumScience and Engineering vol 80 no 1 pp 32ndash40 2011
[6] S Ji C Tian C Shi J Ye Z Zhang and X Fu ldquoNewunderstanding on water-oil displacement efficiency in a highwater-cut stagerdquo Petroleum Exploration and Development vol39 no 3 pp 362ndash370 2012
6 Journal of Applied Mathematics
[7] F Doster and R Hilfer ldquoGeneralized Buckley-Leverett theoryfor two-phase flow in porous mediardquo New Journal of Physicsvol 13 Article ID 123030 2011
[8] J Wang P Yang and Q Liu ldquoDetermination of relativepermeability curves using data measured with rate-controlledmercury penetrationrdquo Journal of the University of PetroleumChina vol 27 no 4 pp 66ndash69 2003
[9] E DiBenedetto U Gianazza and V Vespri ldquoContinuity of thesaturation in the flow of two immiscible fluids in a porousmediumrdquo Indiana University Mathematics Journal vol 59 no6 pp 2041ndash2076 2010
[10] S S Al-Otaibi and A A Al-Majed ldquoFactors affecting pseudorelative permeability curvesrdquo Journal of Petroleum Science andEngineering vol 21 no 3-4 pp 249ndash261 1998
[11] W Littmann and K Littmann ldquoUnstructured grids for numer-ical reservoir simulationmdashusing TOUGH2 for gas storagerdquo OilGas-European Magazine vol 38 pp 204ndash209 2012
[12] I Brailovsky A Babchin M Frankel and G SivashinskyldquoFingering instability in water-oil displacementrdquo Transport inPorous Media vol 63 no 3 pp 363ndash380 2006
[13] D Bonnery F J Breidt and F Coquet ldquoUniform convergenceof the empirical cumulative distribution function under infor-mative selection from a finite populationrdquo Bernoulli vol 18 no4 pp 1361ndash1385 2012
[14] T Ramstad N Idowu C Nardi and P E Oslashren ldquoRelativepermeability calculations from two-phase flow simulationsdirectly on digital images of porous rocksrdquo Transport in PorousMedia vol 94 no 2 pp 487ndash504 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
1234
n
234m
The average partition total m
The p
artit
ion
tota
l n
Supply edge Fluid producing edge
middot middot middot
re
Figure 1 Reservoir profile model
119870119903119908
(119878119908) = 119886119908(
119878119908minus 119878119908119888
1 minus 119878119908119888
minus 119878119900119903
)
119887119908
119870119903119900(119878119908) = 119886119900(1 minus 119878119900119903minus 119878119908
1 minus 119878119908119888
minus 119878119900119903
)
119887119900
if 119903119890gt 119903119908119891
119894
119895 then 119878
119908
119894(119905)
119895= 119878119868119908119895
if 119903119890le 119903119908119891
119894
119895 then 119878
119908
119894(119905)
119895= 119891minus1
119908
1015840
(119878119908)
119891119908= 119891119908(119878119908)
mathematical model
119870119903119900119894
119895= 119870119903119900 (119878
119908
119894(119905)
119895) 119870119903119908
119894
119895= 119870119903119908 (119878
119908
119894(119905)
119895)
119879119894
119895=
2120587ℎ119895119870119895
ln (119903119903119894119895)(119870119903119900
119894
119895
120583119900
+119870119903119908
119894
119895
120583119908
)
119879119895=
1
sum119898
119894=1(1119879119894119895) 119879 =
119899
sum119895=1
119879119895 119876
119905
119895=119879119895
119879sdot 119876119905
1199032
119890minus 119903119894
119895
2
=119891119905
119908(119878119908
119894
119895)1015840
119876119905
119895
120601119895sdot 120587ℎ119895
119903119908119891
2
119895= 1199032
119890minus119891119905
119908(119878119908119891
119894
119895)1015840
119876119905
119895
120601119895sdot 120587ℎ119895
119876119905
119908119895= 119876119905
119895sdot 119891119905
119908(1198781
119908119895) 119876
119905
119900119895= 119876119905
119895sdot (1 minus 119891
119905
119908(1198781
119908119895))
119876119905
119908=
119899
sum119895=1
119876119905
119908119895 119876
119905
119900=
119899
sum119895=1
119876119905
119900119895 119891
119908(119905) =
119876119905
119908
119876119905119908+ 119876119905119900
(1)
If we know the equations 119870119903119908(119878119908) 119870119903119900(119878119908) another
inverse problem model will be obtained as shown in (2)The second question an optimization distribution prob-
lem 1198701 1198702 119870
119899 and the standard deviation 120590
The inverse problem mathematical model of the forma-tion permeability is as follows
objective function
119864 = min119899119905
sum119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2
where 119899119905 is time steps
initial condition
119878119908
119894(0)
119895= 119878119908119888119895
1198760
119895= 0
119891 (119870119895) =
1
radic2120587 sdot 120590119890minus(ln119870
119895minus120583)2
21205902
120583 = Ln119870
if 119903119890gt 119903119908119891
119894
119895 then 119878
119908
119894(119905)
119895= 119878119908119888119895
if 119903119890le 119903119908119891
119894
119895 then 119878
119908
119894(119905)
119895= 119891minus1
119908
1015840
(119878119908)
119891119908= 119891119908(119878119908)
mathematical model
119870119903119900119894
119895= 119870119903119900 (119878
119908
119894(119905)
119895) 119870119903119908
119894
119895= 119870119903119908 (119878
119908
119894(119905)
119895)
119879119894
119895=
2120587ℎ119895119870119895
ln (119903119903119894119895)(119870119903119900
119894
119895
120583119900
+119870119903119908
119894
119895
120583119908
)
119879119895=
1
sum119898
119894=1(1119879119894119895) 119879 =
119899
sum119895=1
119879119895 119876
119905
119895=119879119895
119879sdot 119876119905
1199032
119890minus 119903119894
119895
2
=119891119905
119908(119878119908
119894
119895)1015840
119876119905
119895
120601119895sdot 120587ℎ119895
119903119908119891
2
119895= 1199032
119890minus119891119905
119908(119878119908119891
119894
119895)1015840
119876119905
119895
120601119895sdot 120587ℎ119895
119876119905
119908119895= 119876119905
119895sdot 119891119905
119908(1198781
119908119895) 119876
119905
119900119895= 119876119905
119895sdot (1 minus 119891
119905
119908(1198781
119908119895))
119876119905
119908=
119899
sum119895=1
119876119905
119908119895 119876
119905
119900=
119899
sum119895=1
119876119905
119900119895 119891
119908(119905) =
119876119905
119908
119876119905119908+ 119876119905119900
(2)
Here 119891(history)119908
(119905) is history water cut and 119891119908(119905) is cal-
culation water cut the first water cut is a known amountfrom production and another is calculated by our inverseproblem models Water cut means water production rate is akey parameter in reservoir engineering which can representwater production capacity in reservoir development If watercut is too high it may bring negative effects to reservoirproduction Therefore the history matching for water pro-duction rate plays an important role in reservoir dynamicanalysis and numerical simulation [4ndash6]
And thus 119894 = 1 2 3 119898 119895 = 1 2 3 119899 119898
is the average partition total 119899 is the partition total 119879is grid conductivity ℎ
119895is each longitudinal layer stratum
Journal of Applied Mathematics 3
dr
Supp
ly ed
ge
Fluid producing edge
h
re
Figure 2 Radial flow unit model
thickness 119870119895is each longitudinal layer permeability 120601
119895is
each longitudinal layer porosity 119878119894119908119895
is grid water saturationat different time 119878
119908119888is the initial water saturation 119864 is
the objective function 119903119890is reservoir radius 119870 is average
permeability 119870119903119900
is oil relative permeability 119870119903119908
is waterrelative permeability119876
119905is the total fluid production119876119905
119908is the
total water yield of fluid producing edge at different time 119876119905119900
is the total oil production of fluid producing edge at differenttime 119876119905
119895is the total fluid production of each longitudinal
layer at different time 119878119908119891
is the frontier saturation 119903119908119891
is thecorresponding displacement of frontier saturation
The inverse models (1) and (2) contain Buckley-Leveretttheory of two-phase flow [7] and establish the oil-water two-phase plane radial flow function 1199032
119890minus 1199032= 1198911015840
119908sdot 119876119905120601 sdot 120587 sdot ℎ
without considering these factors of gravity and capillarypressure in addition that shows the movement rules ofisosaturation surface As said above most researchers alwaysrely on forward problems [8] of core sampling experimentand mathematical statistics for an answer As a result thereis always a large error between the calculation water cut andthe history water cut in reservoir water cut analysis In thispaper according to historical water cut and calculation watercut to establish the objective function 119864 = minsum119899119905
119905=1(119891119908(119905) minus
119891(history)119908
(119905))2 By the end an inverse problem model on the
optimal solution distribution 119886119908 119887119908 119886119900 119887119900 and oil-water
relative permeability equations 119870119903119908(119878119908) 119870119903119900(119878119908) based on
the Buckley-Leveret theory of two-phase flow can be realizedand another inverse problem model was modeled under thecondition of the formation permeability logarithmic functionare always obey normal distribution Finally it can provide akey information for reservoir numerical simulation studies
2 Mathematical Model of Oil-WaterTwo-Phase Plane Radial Flow
Theorem 1 Without considering these factors of gravity andcapillary pressure through the Buckley-Leveret theory of two-phase flow one established the oil-water two-phase plane radialflow mathematical model
1199032
119890minus 1199032=1198911015840
119908sdot 119876119905
120601 sdot 120587ℎ (3)
Proof We suppose the liquid flow rule is a plane radial flowfrom reservoir limit to well and choose a volume element inthe vertical direction of streamline as shown in Figure 2
According to the seepage principle [9] we can get a flowequation of the volume element
119902119908= 119902 sdot 119891
119908 (4)
In the 119889119905 time the flow volume of the volume element is
119876out = 119889119902119908sdot 119889119905 (5)
We can derive from (4) and (5) that
119876out = 119902 sdot 119889119891119908sdot 119889119905 (6)
Meanwhile the inflow volume of the volume element is
119876in = 120601 sdot 2120587ℎ sdot 119903119889119903 sdot 119889119904119908 (7)
In the 119889119905 time relying on the Buckley-Leveret theory oftwo-phase flow 119876in = 119876out from (6) and (7) we have that119902 sdot 119889119891119908sdot 119889119905 = 120601 sdot 2120587ℎ sdot 119903119889119903 sdot 119889119878119908 as follows
119902 sdot119889119891119908
119889119878119908sdot 119889119905 = 120601 sdot 2120587ℎ sdot 119903119889119903 (8)
From 119891119908= 119891119908(119878119908) we can get that 1198911015840
119908= 119889119891119908119889119878119908
Then deriving from (8) that 119902 sdot119889119891119908(119878119908) sdot119889119905 = 120601 sdot2120587ℎ sdot119903119889119903
after infinitesimal calculus we obtain
1198911015840
119908int119905
0
119902 sdot 119889119905 = 120601 sdot 2120587ℎ sdot int119903119890
119903
119903119889119903 (9)
If119876119905 = int119905
0119902 sdot119889119905 then (9) turns to1198911015840
119908sdot119876119905= 120601sdot120587ℎ sdot (119903
2
119890minus1199032)
We obtain
1199032
119890minus 1199032=1198911015840
119908sdot 119876119905
120601 sdot 120587ℎ (10)
From the reservoir profile grid model the plane radial flowmodel of the grids in the longitudinal each layer can showsthat
1199032
119890minus 119903119894
119895
2
=119891119905
119908(119878119908
119894
119895)1015840
119876119905
119895
120601119895sdot 120587ℎ119895
(11)
Theorem 2 A special saturation definition about ldquo119878119908119891rdquo from
the water cut function curve 119891119908
= 119891119908(119878119908) selecting the
point of the irreducible water saturation as a fixed point andjoining any other point on the curve and constructing function119896(119878119908119891) = (119891
119908(119878119908119891)minus119891119908(1198781199081))(119878119908119891minus1198781199081) ifMax119896(119878
119908119891) then
one can call ldquo119878119908119891rdquo frontier saturation Now the corresponding
displacement of frontier saturation 119903119908119891
is
119903119908119891
2
119895= 1199032
119890minus119891119905
119908(119878119908119891
119894
119895)1015840
119876119905
119895
120601119895sdot 120587ℎ119895
(12)
4 Journal of Applied Mathematics
3 The Inverse Problem Model on Oil-WaterRelative Permeability
31 The Oil-Water Relative Permeability Equations In thereservoir engineering the different lithological character hasthe different corresponding oil-water relative permeabilitycurve equations [10] but the most commonly form is usedas a kind of the exponential form expression as follows
119870119903119908
(119878119908) = 119886119908(
119878119908minus 119878119908119888
1 minus 119878119908119888
minus 119878119900119903
)
119887119908
119870119903119900(119878119908) = 119886119900(1 minus 119878119900119903minus 119878119908
1 minus 119878119908119888
minus 119878119900119903
)
119887119900
(13)
In reservoir profile grid model the oil-water relative perme-ability curve equations of each grid can be showed
119870119903119908
(119878119894
119908119895) = 119886119908(
119878119894
119908119895minus 119878119908119888
1 minus 119878119908119888
minus 119878119900119903
)
119887119908
119870119903119900(119878119894
119908119895) = 119886119900(1 minus 119878119900119903minus 119878119894
119908119895
1 minus 119878119908119888
minus 119878119900119903
)
119887119900
(14)
In the equations we can know different 119886119908 119887119908 119886119900 119887119900 corre-
sponding to its own 119870119903119908(119878119908) 119870119903119900(119878119908)
32 The Inverse Problem Mathematical Model of Oil-WaterRelative Permeability Equations
Theorem 3 If the oil-water relative permeability equations119870119903119908(119878119908) 119870119903119900(119878119908) always obey (13) and one can calculate the
water cut 119891119908(119905) based on the model of the Theorem 1 if the
objective function 119864 = minsum119899119905119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2 can
be satisfied then the inverse problem model (1) will have theoptimal solution of the distribution 119886
119908 119887119908 119886119900 119887119900 and oil-
water relative permeability equations 119870119903119908(119878119908) 119870119903119900(119878119908)
Proof Relying on oil-water relative permeability equations(14) and the conductivity definition of numerical reservoirsimulation [11] we get the following
initial value 1198860
119908 1198870
119908 1198860
119900 1198870
119900
119870119903119908
(119878119908) = 1198860
119908(
119878119908minus 119878119908119888
1 minus 119878119908119888
minus 119878119900119903
)
1198870
119908
119870119903119900(119878119908) = 1198860
119900(1 minus 119878119900119903minus 119878119908
1 minus 119878119908119888
minus 119878119900119903
)
1198870
119900
119879119894
119895=
2120587ℎ119895119870119895
ln (119903119903119894119895)(119870119903119900
119894
119895
120583119900
+119870119903119908
119894
119895
120583119908
)
119879119895=
1
sum119898
119894=1(1119879119894119895) 119879 =
119899
sum119895=1
119879119895
119876119905
119895=119879119895
119879sdot 119876119905
(15)
We can derive from (12) and (15) 119903119908119891
If 119903119890gt 119903119908119891 then 119878
119908
119894(119905)
119895= 119878119908119888119895
If 119903119890le 119903119908119891
solved inversefunction of (11) and calculated water saturation of the fluidproducing edge then 119878
119908
119894(119905)
119895= 119891minus1
119908
1015840
(119878119908) and so forth 119894 = 1
If we obtain the value of the water saturation [12] relyingon the function 119891
119908(119878119908) sim 119878
119908 we can calculate the value
of the longitudinal each layer water cut 119891119905119908(1198781
119908119895) and the
constructing mathematic model as follows
119876119905
119908119895= 119876119905
119895sdot 119891119905
119908(1198781
119908119895)
119876119905
119900119895= 119876119905
119895sdot (1 minus 119891
119905
119908(1198781
119908119895))
119876119905
119908=
119899
sum119895=1
119876119905
119908119895 119876
119905
119900=
119899
sum119895=1
119876119905
119900119895
119891119908(119905) =
119876119905
119908
119876119905119908+ 119876119905119900
(16)
From (16) we can calculate the total water cut of the fluidproducing edge 119891
119908(119905) and rely on the objective function 119864 =
minsum119899119905119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2 with numerical optimization
calculation in the inverse problem model an optimizationproblem of the distribution 119886
119908 119887119908 119886119900 119887119900 will be obtained
so the 119870119903119908(119878119908) and 119870
119903119900(119878119908) can be formed The proof of
Theorem 3 is completed
4 The Inverse Problem Model on theFormation Permeability LogarithmicFunction Always Obey Normal Distribution
Theorem 4 If the formation permeability logarithmic func-tion distribution Ln119870
1 Ln119870
2 Ln119870
119899 always obey nor-
mal distribution 119891(119870) = (1radic2120587 sdot 120590)119890minus(ln119870minus120583)221205902 120583 =
Ln119870 meanwhile one calculates the value of the 119891119908(119905) based
on the model of Theorem 1 which can be adapted to theobjective function 119864 = minsum119899119905
119905=1(119891119908(119905)minus119891
(history)119908
(119905))2 then the
inverse problem model (2) will have the optimal distributionLn119870
1 Ln119870
2 Ln119870
119899 and 119870
1 1198702 119870
119899
Proof According to 3120590 principle for normal distributioncurve if 119875(120583minus3120590minus119886 lt 119883 le 120583+3120590+119886) rarr 1 then there are aleft end point value and a right end point value on the curvethe119883 value of the left end point is ln119870min or 120583 minus 3120590 minus 119886 The119883 value of the right end point is ln119870max or 120583 + 3120590 + 119886 Andthus 119886 gt 0 and 120583 = ln119870 = 05 sdot (ln119870min + ln119870max)
According to the area superposition principle of normaldistribution curve Select 119909 = ln119870 as the starting point stepsize Δ119909 and make a subdivision for probability curve if thearea summation of these formed closed figures can infinitelyapproach 1 then we can obtain the ln119870min value of the leftend point and the ln119870max value of the right end point andln119870max = 2 sdot ln119870 minus ln119870min
According to the area superposition principle of normaldistribution curve and giving a initial value 120590 of normaldistribution then119883
0= ln119870min and119883119899 = ln119870max 119865(119883119895) is a
cumulative distribution function of the normal distribution[13] If we use equal step size Δ119883 make a subdivision for
Journal of Applied Mathematics 5
probability curve then the area summation of every closedfigures 119878
119895= 119865(119883
119895) minus119865(119883
119895minus1) each longitudinal layer stratum
thickness ℎ119895= ℎ sdot 119878
119895 and 119883
119895= Δ119883 sdot 119895 + 119883
0 119870119895= 119890119883119895
if we use equal area Δ119878 make a subdivision for probabilitycurve then the area summation of every closed figures 119878
119895=
119865(119883119895)minus119865(119883
119895minus1) 119878119895= Δ119878 = 1119899 ℎ
119895= ℎsdot119878
119895 and119883
119895= 119865minus1(119878119895)
119870119895= 119890119883119895 And thus 119895 = 1 2 3 119899 (119899 is the total number of
vertical stratification)Relying on the distribution 119870
1 1198702 119870
119899 oil-water
relative permeability functions 119896119903119900 = 119896119903119900(119878119908) 119896119903119908 =
119896119903119908(119878119908) [14] the conductivity definition of numerical reser-
voir simulation we get the following equations
119896119903119900119894
119895= 119896119903119900 (119878
119908
119894(119905)
119895) 119896119903119908
119894
119895= 119896119903119908 (119878
119908
119894(119905)
119895)
119879119894
119895=
2120587ℎ119895119870119895
ln (119903119903119894119895)(119870119903119900
119894
119895
120583119900
+119870119903119908
119894
119895
120583119908
)
119879119895=
1
sum119898
119894=1(1119879119894119895) 119879 =
119899
sum119895=1
119879119895
119876119905
119895=119879119895
119879sdot 119876119905
(17)
We can derive from (12) and (17) that 119903119908119891119895
If 119903119890
le 119903119908119891119895
solving inverse function of (11) andcalculating water saturation of the fluid producing edge then119878119908
119894(119905)
119895= 119865minus1(119903119894
119895 120601119895 ℎ119895 119876119905
119895) and thus 119894 = 1 If 119903
119890gt 119903119908119891119895
then
119878119908
119894(119905)
119895= 119878119908119888119895
and thus 119894 = 1If we obtain the value of the water saturation (119878
119908) relying
on the function119891119908(119878119908) sim 119878119908 we can calculate the value of the
longitudinal each layer water cut 119891119905119908(1198781
119908119895) then mathematic
model can be constructed as follows
119876119905
119908119895= 119876119905
119895sdot 119891119905
119908(1198781
119908119895) 119876
119905
119900119895= 119876119905
119895sdot (1 minus 119891
119905
119908(1198781
119908119895))
119876119905
119908=
119899
sum119895=1
119876119905
119908119895 119876
119905
119900=
119899
sum119895=1
119876119905
119900119895
119891119908(119905) =
119876119905
119908
(119876119905119908+ 119876119905119900)
(18)
Equation (18) can calculate the total water cut of the fluidproducing edge 119891
119908(119905) and relies on the objective function
119864 = minsum119899119905119905=1
(119891119908(119905)minus119891
(history)119908
(119905))2With numerical optimiza-
tion calculation in the inverse problem model the standarddeviation 120590 and an optimization problem of the distributionLn1198701 Ln1198702 Ln119870
119899 will be obtained so the distribution
1198701 1198702 119870
119899 will be given The proof of Theorem 4 is
completed
5 Discussions and Conclusions
The different distribution 119886119908 119887119908 119886119900 119887119900 corresponds to a
group of oil-water relative permeability equations 119870119903119908(119878119908)
and 119870119903119900(119878119908) Combining with the objective function
119864 = minsum119899119905119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2 through the optimization
solution method it will finally bring a set of the optimumdistribution 119886
119908 119887119908 119886119900 119887119900 and the oil-water relative
permeability equations119870119903119908(119878119908) and119870
119903119900(119878119908)
According to the above inverse problem mathematicalmodel based on the definition of the normal distributiondifferent (120583
119894 1205902
119894) corresponds to its own normal distribution
curve with the certain expectation 120583119894 If 120590
119894goes up the
volatility of its normal distribution will be stronger accordingto the definition of standard deviation which will finally leadto the large differential permeability distribution namelythe strong heterogeneity With the certain expectation 120583
119894
each different value of 120590119894corresponds to a group of original
values of (120583119894 1205902
119894) which will yield a group of values of
1198701 1198702 119870
119899minus1 119870119899 namely the value of permeability of
each single formation And it will also work out liquidproduction water production and integrated water cut of thewhole liquid outlet of each single formation Combining withthe objective function 119864 = minsum119899119905
119905=1(119891119908(119905) minus 119891
(history)119908
(119905))2
it will finally bring a set of optimum normal distributionsof 119883 sim (120583
119894 1205902
119894) (Etc 119883 = ln119870) through the optimization
solution method Analysis of the changes of water drivingplace oil production and water production can be donethrough the related mathematical model
Finally the idea of constructing inverse problem modelsaccording to the historical dynamic production data canbe realized which attaches great importance to formationheterogeneity observation of water flooding front positionand prediction of dynamic producing performance
Acknowledgments
The authors gratefully thank the support of Canada CMGFoundation and the referees for valuable suggestions
References
[1] S X Bing ldquoReservoir numerical simulation study based onformation parametersrsquo time-variabilityrdquo Procedia Engineeringvol 29 pp 2327ndash2331 2012
[2] F A Dumkwu A W Islam and E S Carlson ldquoReview ofwell models and assessment of their impacts on numericalreservoir simulation performancerdquo Journal of Petroleum Scienceand Engineering vol 82-83 pp 174ndash186 2012
[3] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011
[4] S Haj-shafiei S Ghosh and D Rousseau ldquoKinetic stability andrheology of wax-stabilized water-in-oil emulsions at differentwater cutsrdquo Journal of Colloid and Interface Science vol 410 pp11ndash20 2013
[5] S Talatori and T Barth ldquoRate of hydrate formation in crudeoilgaswater emulsions with different water cutsrdquo Journal ofPetroleumScience and Engineering vol 80 no 1 pp 32ndash40 2011
[6] S Ji C Tian C Shi J Ye Z Zhang and X Fu ldquoNewunderstanding on water-oil displacement efficiency in a highwater-cut stagerdquo Petroleum Exploration and Development vol39 no 3 pp 362ndash370 2012
6 Journal of Applied Mathematics
[7] F Doster and R Hilfer ldquoGeneralized Buckley-Leverett theoryfor two-phase flow in porous mediardquo New Journal of Physicsvol 13 Article ID 123030 2011
[8] J Wang P Yang and Q Liu ldquoDetermination of relativepermeability curves using data measured with rate-controlledmercury penetrationrdquo Journal of the University of PetroleumChina vol 27 no 4 pp 66ndash69 2003
[9] E DiBenedetto U Gianazza and V Vespri ldquoContinuity of thesaturation in the flow of two immiscible fluids in a porousmediumrdquo Indiana University Mathematics Journal vol 59 no6 pp 2041ndash2076 2010
[10] S S Al-Otaibi and A A Al-Majed ldquoFactors affecting pseudorelative permeability curvesrdquo Journal of Petroleum Science andEngineering vol 21 no 3-4 pp 249ndash261 1998
[11] W Littmann and K Littmann ldquoUnstructured grids for numer-ical reservoir simulationmdashusing TOUGH2 for gas storagerdquo OilGas-European Magazine vol 38 pp 204ndash209 2012
[12] I Brailovsky A Babchin M Frankel and G SivashinskyldquoFingering instability in water-oil displacementrdquo Transport inPorous Media vol 63 no 3 pp 363ndash380 2006
[13] D Bonnery F J Breidt and F Coquet ldquoUniform convergenceof the empirical cumulative distribution function under infor-mative selection from a finite populationrdquo Bernoulli vol 18 no4 pp 1361ndash1385 2012
[14] T Ramstad N Idowu C Nardi and P E Oslashren ldquoRelativepermeability calculations from two-phase flow simulationsdirectly on digital images of porous rocksrdquo Transport in PorousMedia vol 94 no 2 pp 487ndash504 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
dr
Supp
ly ed
ge
Fluid producing edge
h
re
Figure 2 Radial flow unit model
thickness 119870119895is each longitudinal layer permeability 120601
119895is
each longitudinal layer porosity 119878119894119908119895
is grid water saturationat different time 119878
119908119888is the initial water saturation 119864 is
the objective function 119903119890is reservoir radius 119870 is average
permeability 119870119903119900
is oil relative permeability 119870119903119908
is waterrelative permeability119876
119905is the total fluid production119876119905
119908is the
total water yield of fluid producing edge at different time 119876119905119900
is the total oil production of fluid producing edge at differenttime 119876119905
119895is the total fluid production of each longitudinal
layer at different time 119878119908119891
is the frontier saturation 119903119908119891
is thecorresponding displacement of frontier saturation
The inverse models (1) and (2) contain Buckley-Leveretttheory of two-phase flow [7] and establish the oil-water two-phase plane radial flow function 1199032
119890minus 1199032= 1198911015840
119908sdot 119876119905120601 sdot 120587 sdot ℎ
without considering these factors of gravity and capillarypressure in addition that shows the movement rules ofisosaturation surface As said above most researchers alwaysrely on forward problems [8] of core sampling experimentand mathematical statistics for an answer As a result thereis always a large error between the calculation water cut andthe history water cut in reservoir water cut analysis In thispaper according to historical water cut and calculation watercut to establish the objective function 119864 = minsum119899119905
119905=1(119891119908(119905) minus
119891(history)119908
(119905))2 By the end an inverse problem model on the
optimal solution distribution 119886119908 119887119908 119886119900 119887119900 and oil-water
relative permeability equations 119870119903119908(119878119908) 119870119903119900(119878119908) based on
the Buckley-Leveret theory of two-phase flow can be realizedand another inverse problem model was modeled under thecondition of the formation permeability logarithmic functionare always obey normal distribution Finally it can provide akey information for reservoir numerical simulation studies
2 Mathematical Model of Oil-WaterTwo-Phase Plane Radial Flow
Theorem 1 Without considering these factors of gravity andcapillary pressure through the Buckley-Leveret theory of two-phase flow one established the oil-water two-phase plane radialflow mathematical model
1199032
119890minus 1199032=1198911015840
119908sdot 119876119905
120601 sdot 120587ℎ (3)
Proof We suppose the liquid flow rule is a plane radial flowfrom reservoir limit to well and choose a volume element inthe vertical direction of streamline as shown in Figure 2
According to the seepage principle [9] we can get a flowequation of the volume element
119902119908= 119902 sdot 119891
119908 (4)
In the 119889119905 time the flow volume of the volume element is
119876out = 119889119902119908sdot 119889119905 (5)
We can derive from (4) and (5) that
119876out = 119902 sdot 119889119891119908sdot 119889119905 (6)
Meanwhile the inflow volume of the volume element is
119876in = 120601 sdot 2120587ℎ sdot 119903119889119903 sdot 119889119904119908 (7)
In the 119889119905 time relying on the Buckley-Leveret theory oftwo-phase flow 119876in = 119876out from (6) and (7) we have that119902 sdot 119889119891119908sdot 119889119905 = 120601 sdot 2120587ℎ sdot 119903119889119903 sdot 119889119878119908 as follows
119902 sdot119889119891119908
119889119878119908sdot 119889119905 = 120601 sdot 2120587ℎ sdot 119903119889119903 (8)
From 119891119908= 119891119908(119878119908) we can get that 1198911015840
119908= 119889119891119908119889119878119908
Then deriving from (8) that 119902 sdot119889119891119908(119878119908) sdot119889119905 = 120601 sdot2120587ℎ sdot119903119889119903
after infinitesimal calculus we obtain
1198911015840
119908int119905
0
119902 sdot 119889119905 = 120601 sdot 2120587ℎ sdot int119903119890
119903
119903119889119903 (9)
If119876119905 = int119905
0119902 sdot119889119905 then (9) turns to1198911015840
119908sdot119876119905= 120601sdot120587ℎ sdot (119903
2
119890minus1199032)
We obtain
1199032
119890minus 1199032=1198911015840
119908sdot 119876119905
120601 sdot 120587ℎ (10)
From the reservoir profile grid model the plane radial flowmodel of the grids in the longitudinal each layer can showsthat
1199032
119890minus 119903119894
119895
2
=119891119905
119908(119878119908
119894
119895)1015840
119876119905
119895
120601119895sdot 120587ℎ119895
(11)
Theorem 2 A special saturation definition about ldquo119878119908119891rdquo from
the water cut function curve 119891119908
= 119891119908(119878119908) selecting the
point of the irreducible water saturation as a fixed point andjoining any other point on the curve and constructing function119896(119878119908119891) = (119891
119908(119878119908119891)minus119891119908(1198781199081))(119878119908119891minus1198781199081) ifMax119896(119878
119908119891) then
one can call ldquo119878119908119891rdquo frontier saturation Now the corresponding
displacement of frontier saturation 119903119908119891
is
119903119908119891
2
119895= 1199032
119890minus119891119905
119908(119878119908119891
119894
119895)1015840
119876119905
119895
120601119895sdot 120587ℎ119895
(12)
4 Journal of Applied Mathematics
3 The Inverse Problem Model on Oil-WaterRelative Permeability
31 The Oil-Water Relative Permeability Equations In thereservoir engineering the different lithological character hasthe different corresponding oil-water relative permeabilitycurve equations [10] but the most commonly form is usedas a kind of the exponential form expression as follows
119870119903119908
(119878119908) = 119886119908(
119878119908minus 119878119908119888
1 minus 119878119908119888
minus 119878119900119903
)
119887119908
119870119903119900(119878119908) = 119886119900(1 minus 119878119900119903minus 119878119908
1 minus 119878119908119888
minus 119878119900119903
)
119887119900
(13)
In reservoir profile grid model the oil-water relative perme-ability curve equations of each grid can be showed
119870119903119908
(119878119894
119908119895) = 119886119908(
119878119894
119908119895minus 119878119908119888
1 minus 119878119908119888
minus 119878119900119903
)
119887119908
119870119903119900(119878119894
119908119895) = 119886119900(1 minus 119878119900119903minus 119878119894
119908119895
1 minus 119878119908119888
minus 119878119900119903
)
119887119900
(14)
In the equations we can know different 119886119908 119887119908 119886119900 119887119900 corre-
sponding to its own 119870119903119908(119878119908) 119870119903119900(119878119908)
32 The Inverse Problem Mathematical Model of Oil-WaterRelative Permeability Equations
Theorem 3 If the oil-water relative permeability equations119870119903119908(119878119908) 119870119903119900(119878119908) always obey (13) and one can calculate the
water cut 119891119908(119905) based on the model of the Theorem 1 if the
objective function 119864 = minsum119899119905119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2 can
be satisfied then the inverse problem model (1) will have theoptimal solution of the distribution 119886
119908 119887119908 119886119900 119887119900 and oil-
water relative permeability equations 119870119903119908(119878119908) 119870119903119900(119878119908)
Proof Relying on oil-water relative permeability equations(14) and the conductivity definition of numerical reservoirsimulation [11] we get the following
initial value 1198860
119908 1198870
119908 1198860
119900 1198870
119900
119870119903119908
(119878119908) = 1198860
119908(
119878119908minus 119878119908119888
1 minus 119878119908119888
minus 119878119900119903
)
1198870
119908
119870119903119900(119878119908) = 1198860
119900(1 minus 119878119900119903minus 119878119908
1 minus 119878119908119888
minus 119878119900119903
)
1198870
119900
119879119894
119895=
2120587ℎ119895119870119895
ln (119903119903119894119895)(119870119903119900
119894
119895
120583119900
+119870119903119908
119894
119895
120583119908
)
119879119895=
1
sum119898
119894=1(1119879119894119895) 119879 =
119899
sum119895=1
119879119895
119876119905
119895=119879119895
119879sdot 119876119905
(15)
We can derive from (12) and (15) 119903119908119891
If 119903119890gt 119903119908119891 then 119878
119908
119894(119905)
119895= 119878119908119888119895
If 119903119890le 119903119908119891
solved inversefunction of (11) and calculated water saturation of the fluidproducing edge then 119878
119908
119894(119905)
119895= 119891minus1
119908
1015840
(119878119908) and so forth 119894 = 1
If we obtain the value of the water saturation [12] relyingon the function 119891
119908(119878119908) sim 119878
119908 we can calculate the value
of the longitudinal each layer water cut 119891119905119908(1198781
119908119895) and the
constructing mathematic model as follows
119876119905
119908119895= 119876119905
119895sdot 119891119905
119908(1198781
119908119895)
119876119905
119900119895= 119876119905
119895sdot (1 minus 119891
119905
119908(1198781
119908119895))
119876119905
119908=
119899
sum119895=1
119876119905
119908119895 119876
119905
119900=
119899
sum119895=1
119876119905
119900119895
119891119908(119905) =
119876119905
119908
119876119905119908+ 119876119905119900
(16)
From (16) we can calculate the total water cut of the fluidproducing edge 119891
119908(119905) and rely on the objective function 119864 =
minsum119899119905119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2 with numerical optimization
calculation in the inverse problem model an optimizationproblem of the distribution 119886
119908 119887119908 119886119900 119887119900 will be obtained
so the 119870119903119908(119878119908) and 119870
119903119900(119878119908) can be formed The proof of
Theorem 3 is completed
4 The Inverse Problem Model on theFormation Permeability LogarithmicFunction Always Obey Normal Distribution
Theorem 4 If the formation permeability logarithmic func-tion distribution Ln119870
1 Ln119870
2 Ln119870
119899 always obey nor-
mal distribution 119891(119870) = (1radic2120587 sdot 120590)119890minus(ln119870minus120583)221205902 120583 =
Ln119870 meanwhile one calculates the value of the 119891119908(119905) based
on the model of Theorem 1 which can be adapted to theobjective function 119864 = minsum119899119905
119905=1(119891119908(119905)minus119891
(history)119908
(119905))2 then the
inverse problem model (2) will have the optimal distributionLn119870
1 Ln119870
2 Ln119870
119899 and 119870
1 1198702 119870
119899
Proof According to 3120590 principle for normal distributioncurve if 119875(120583minus3120590minus119886 lt 119883 le 120583+3120590+119886) rarr 1 then there are aleft end point value and a right end point value on the curvethe119883 value of the left end point is ln119870min or 120583 minus 3120590 minus 119886 The119883 value of the right end point is ln119870max or 120583 + 3120590 + 119886 Andthus 119886 gt 0 and 120583 = ln119870 = 05 sdot (ln119870min + ln119870max)
According to the area superposition principle of normaldistribution curve Select 119909 = ln119870 as the starting point stepsize Δ119909 and make a subdivision for probability curve if thearea summation of these formed closed figures can infinitelyapproach 1 then we can obtain the ln119870min value of the leftend point and the ln119870max value of the right end point andln119870max = 2 sdot ln119870 minus ln119870min
According to the area superposition principle of normaldistribution curve and giving a initial value 120590 of normaldistribution then119883
0= ln119870min and119883119899 = ln119870max 119865(119883119895) is a
cumulative distribution function of the normal distribution[13] If we use equal step size Δ119883 make a subdivision for
Journal of Applied Mathematics 5
probability curve then the area summation of every closedfigures 119878
119895= 119865(119883
119895) minus119865(119883
119895minus1) each longitudinal layer stratum
thickness ℎ119895= ℎ sdot 119878
119895 and 119883
119895= Δ119883 sdot 119895 + 119883
0 119870119895= 119890119883119895
if we use equal area Δ119878 make a subdivision for probabilitycurve then the area summation of every closed figures 119878
119895=
119865(119883119895)minus119865(119883
119895minus1) 119878119895= Δ119878 = 1119899 ℎ
119895= ℎsdot119878
119895 and119883
119895= 119865minus1(119878119895)
119870119895= 119890119883119895 And thus 119895 = 1 2 3 119899 (119899 is the total number of
vertical stratification)Relying on the distribution 119870
1 1198702 119870
119899 oil-water
relative permeability functions 119896119903119900 = 119896119903119900(119878119908) 119896119903119908 =
119896119903119908(119878119908) [14] the conductivity definition of numerical reser-
voir simulation we get the following equations
119896119903119900119894
119895= 119896119903119900 (119878
119908
119894(119905)
119895) 119896119903119908
119894
119895= 119896119903119908 (119878
119908
119894(119905)
119895)
119879119894
119895=
2120587ℎ119895119870119895
ln (119903119903119894119895)(119870119903119900
119894
119895
120583119900
+119870119903119908
119894
119895
120583119908
)
119879119895=
1
sum119898
119894=1(1119879119894119895) 119879 =
119899
sum119895=1
119879119895
119876119905
119895=119879119895
119879sdot 119876119905
(17)
We can derive from (12) and (17) that 119903119908119891119895
If 119903119890
le 119903119908119891119895
solving inverse function of (11) andcalculating water saturation of the fluid producing edge then119878119908
119894(119905)
119895= 119865minus1(119903119894
119895 120601119895 ℎ119895 119876119905
119895) and thus 119894 = 1 If 119903
119890gt 119903119908119891119895
then
119878119908
119894(119905)
119895= 119878119908119888119895
and thus 119894 = 1If we obtain the value of the water saturation (119878
119908) relying
on the function119891119908(119878119908) sim 119878119908 we can calculate the value of the
longitudinal each layer water cut 119891119905119908(1198781
119908119895) then mathematic
model can be constructed as follows
119876119905
119908119895= 119876119905
119895sdot 119891119905
119908(1198781
119908119895) 119876
119905
119900119895= 119876119905
119895sdot (1 minus 119891
119905
119908(1198781
119908119895))
119876119905
119908=
119899
sum119895=1
119876119905
119908119895 119876
119905
119900=
119899
sum119895=1
119876119905
119900119895
119891119908(119905) =
119876119905
119908
(119876119905119908+ 119876119905119900)
(18)
Equation (18) can calculate the total water cut of the fluidproducing edge 119891
119908(119905) and relies on the objective function
119864 = minsum119899119905119905=1
(119891119908(119905)minus119891
(history)119908
(119905))2With numerical optimiza-
tion calculation in the inverse problem model the standarddeviation 120590 and an optimization problem of the distributionLn1198701 Ln1198702 Ln119870
119899 will be obtained so the distribution
1198701 1198702 119870
119899 will be given The proof of Theorem 4 is
completed
5 Discussions and Conclusions
The different distribution 119886119908 119887119908 119886119900 119887119900 corresponds to a
group of oil-water relative permeability equations 119870119903119908(119878119908)
and 119870119903119900(119878119908) Combining with the objective function
119864 = minsum119899119905119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2 through the optimization
solution method it will finally bring a set of the optimumdistribution 119886
119908 119887119908 119886119900 119887119900 and the oil-water relative
permeability equations119870119903119908(119878119908) and119870
119903119900(119878119908)
According to the above inverse problem mathematicalmodel based on the definition of the normal distributiondifferent (120583
119894 1205902
119894) corresponds to its own normal distribution
curve with the certain expectation 120583119894 If 120590
119894goes up the
volatility of its normal distribution will be stronger accordingto the definition of standard deviation which will finally leadto the large differential permeability distribution namelythe strong heterogeneity With the certain expectation 120583
119894
each different value of 120590119894corresponds to a group of original
values of (120583119894 1205902
119894) which will yield a group of values of
1198701 1198702 119870
119899minus1 119870119899 namely the value of permeability of
each single formation And it will also work out liquidproduction water production and integrated water cut of thewhole liquid outlet of each single formation Combining withthe objective function 119864 = minsum119899119905
119905=1(119891119908(119905) minus 119891
(history)119908
(119905))2
it will finally bring a set of optimum normal distributionsof 119883 sim (120583
119894 1205902
119894) (Etc 119883 = ln119870) through the optimization
solution method Analysis of the changes of water drivingplace oil production and water production can be donethrough the related mathematical model
Finally the idea of constructing inverse problem modelsaccording to the historical dynamic production data canbe realized which attaches great importance to formationheterogeneity observation of water flooding front positionand prediction of dynamic producing performance
Acknowledgments
The authors gratefully thank the support of Canada CMGFoundation and the referees for valuable suggestions
References
[1] S X Bing ldquoReservoir numerical simulation study based onformation parametersrsquo time-variabilityrdquo Procedia Engineeringvol 29 pp 2327ndash2331 2012
[2] F A Dumkwu A W Islam and E S Carlson ldquoReview ofwell models and assessment of their impacts on numericalreservoir simulation performancerdquo Journal of Petroleum Scienceand Engineering vol 82-83 pp 174ndash186 2012
[3] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011
[4] S Haj-shafiei S Ghosh and D Rousseau ldquoKinetic stability andrheology of wax-stabilized water-in-oil emulsions at differentwater cutsrdquo Journal of Colloid and Interface Science vol 410 pp11ndash20 2013
[5] S Talatori and T Barth ldquoRate of hydrate formation in crudeoilgaswater emulsions with different water cutsrdquo Journal ofPetroleumScience and Engineering vol 80 no 1 pp 32ndash40 2011
[6] S Ji C Tian C Shi J Ye Z Zhang and X Fu ldquoNewunderstanding on water-oil displacement efficiency in a highwater-cut stagerdquo Petroleum Exploration and Development vol39 no 3 pp 362ndash370 2012
6 Journal of Applied Mathematics
[7] F Doster and R Hilfer ldquoGeneralized Buckley-Leverett theoryfor two-phase flow in porous mediardquo New Journal of Physicsvol 13 Article ID 123030 2011
[8] J Wang P Yang and Q Liu ldquoDetermination of relativepermeability curves using data measured with rate-controlledmercury penetrationrdquo Journal of the University of PetroleumChina vol 27 no 4 pp 66ndash69 2003
[9] E DiBenedetto U Gianazza and V Vespri ldquoContinuity of thesaturation in the flow of two immiscible fluids in a porousmediumrdquo Indiana University Mathematics Journal vol 59 no6 pp 2041ndash2076 2010
[10] S S Al-Otaibi and A A Al-Majed ldquoFactors affecting pseudorelative permeability curvesrdquo Journal of Petroleum Science andEngineering vol 21 no 3-4 pp 249ndash261 1998
[11] W Littmann and K Littmann ldquoUnstructured grids for numer-ical reservoir simulationmdashusing TOUGH2 for gas storagerdquo OilGas-European Magazine vol 38 pp 204ndash209 2012
[12] I Brailovsky A Babchin M Frankel and G SivashinskyldquoFingering instability in water-oil displacementrdquo Transport inPorous Media vol 63 no 3 pp 363ndash380 2006
[13] D Bonnery F J Breidt and F Coquet ldquoUniform convergenceof the empirical cumulative distribution function under infor-mative selection from a finite populationrdquo Bernoulli vol 18 no4 pp 1361ndash1385 2012
[14] T Ramstad N Idowu C Nardi and P E Oslashren ldquoRelativepermeability calculations from two-phase flow simulationsdirectly on digital images of porous rocksrdquo Transport in PorousMedia vol 94 no 2 pp 487ndash504 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
3 The Inverse Problem Model on Oil-WaterRelative Permeability
31 The Oil-Water Relative Permeability Equations In thereservoir engineering the different lithological character hasthe different corresponding oil-water relative permeabilitycurve equations [10] but the most commonly form is usedas a kind of the exponential form expression as follows
119870119903119908
(119878119908) = 119886119908(
119878119908minus 119878119908119888
1 minus 119878119908119888
minus 119878119900119903
)
119887119908
119870119903119900(119878119908) = 119886119900(1 minus 119878119900119903minus 119878119908
1 minus 119878119908119888
minus 119878119900119903
)
119887119900
(13)
In reservoir profile grid model the oil-water relative perme-ability curve equations of each grid can be showed
119870119903119908
(119878119894
119908119895) = 119886119908(
119878119894
119908119895minus 119878119908119888
1 minus 119878119908119888
minus 119878119900119903
)
119887119908
119870119903119900(119878119894
119908119895) = 119886119900(1 minus 119878119900119903minus 119878119894
119908119895
1 minus 119878119908119888
minus 119878119900119903
)
119887119900
(14)
In the equations we can know different 119886119908 119887119908 119886119900 119887119900 corre-
sponding to its own 119870119903119908(119878119908) 119870119903119900(119878119908)
32 The Inverse Problem Mathematical Model of Oil-WaterRelative Permeability Equations
Theorem 3 If the oil-water relative permeability equations119870119903119908(119878119908) 119870119903119900(119878119908) always obey (13) and one can calculate the
water cut 119891119908(119905) based on the model of the Theorem 1 if the
objective function 119864 = minsum119899119905119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2 can
be satisfied then the inverse problem model (1) will have theoptimal solution of the distribution 119886
119908 119887119908 119886119900 119887119900 and oil-
water relative permeability equations 119870119903119908(119878119908) 119870119903119900(119878119908)
Proof Relying on oil-water relative permeability equations(14) and the conductivity definition of numerical reservoirsimulation [11] we get the following
initial value 1198860
119908 1198870
119908 1198860
119900 1198870
119900
119870119903119908
(119878119908) = 1198860
119908(
119878119908minus 119878119908119888
1 minus 119878119908119888
minus 119878119900119903
)
1198870
119908
119870119903119900(119878119908) = 1198860
119900(1 minus 119878119900119903minus 119878119908
1 minus 119878119908119888
minus 119878119900119903
)
1198870
119900
119879119894
119895=
2120587ℎ119895119870119895
ln (119903119903119894119895)(119870119903119900
119894
119895
120583119900
+119870119903119908
119894
119895
120583119908
)
119879119895=
1
sum119898
119894=1(1119879119894119895) 119879 =
119899
sum119895=1
119879119895
119876119905
119895=119879119895
119879sdot 119876119905
(15)
We can derive from (12) and (15) 119903119908119891
If 119903119890gt 119903119908119891 then 119878
119908
119894(119905)
119895= 119878119908119888119895
If 119903119890le 119903119908119891
solved inversefunction of (11) and calculated water saturation of the fluidproducing edge then 119878
119908
119894(119905)
119895= 119891minus1
119908
1015840
(119878119908) and so forth 119894 = 1
If we obtain the value of the water saturation [12] relyingon the function 119891
119908(119878119908) sim 119878
119908 we can calculate the value
of the longitudinal each layer water cut 119891119905119908(1198781
119908119895) and the
constructing mathematic model as follows
119876119905
119908119895= 119876119905
119895sdot 119891119905
119908(1198781
119908119895)
119876119905
119900119895= 119876119905
119895sdot (1 minus 119891
119905
119908(1198781
119908119895))
119876119905
119908=
119899
sum119895=1
119876119905
119908119895 119876
119905
119900=
119899
sum119895=1
119876119905
119900119895
119891119908(119905) =
119876119905
119908
119876119905119908+ 119876119905119900
(16)
From (16) we can calculate the total water cut of the fluidproducing edge 119891
119908(119905) and rely on the objective function 119864 =
minsum119899119905119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2 with numerical optimization
calculation in the inverse problem model an optimizationproblem of the distribution 119886
119908 119887119908 119886119900 119887119900 will be obtained
so the 119870119903119908(119878119908) and 119870
119903119900(119878119908) can be formed The proof of
Theorem 3 is completed
4 The Inverse Problem Model on theFormation Permeability LogarithmicFunction Always Obey Normal Distribution
Theorem 4 If the formation permeability logarithmic func-tion distribution Ln119870
1 Ln119870
2 Ln119870
119899 always obey nor-
mal distribution 119891(119870) = (1radic2120587 sdot 120590)119890minus(ln119870minus120583)221205902 120583 =
Ln119870 meanwhile one calculates the value of the 119891119908(119905) based
on the model of Theorem 1 which can be adapted to theobjective function 119864 = minsum119899119905
119905=1(119891119908(119905)minus119891
(history)119908
(119905))2 then the
inverse problem model (2) will have the optimal distributionLn119870
1 Ln119870
2 Ln119870
119899 and 119870
1 1198702 119870
119899
Proof According to 3120590 principle for normal distributioncurve if 119875(120583minus3120590minus119886 lt 119883 le 120583+3120590+119886) rarr 1 then there are aleft end point value and a right end point value on the curvethe119883 value of the left end point is ln119870min or 120583 minus 3120590 minus 119886 The119883 value of the right end point is ln119870max or 120583 + 3120590 + 119886 Andthus 119886 gt 0 and 120583 = ln119870 = 05 sdot (ln119870min + ln119870max)
According to the area superposition principle of normaldistribution curve Select 119909 = ln119870 as the starting point stepsize Δ119909 and make a subdivision for probability curve if thearea summation of these formed closed figures can infinitelyapproach 1 then we can obtain the ln119870min value of the leftend point and the ln119870max value of the right end point andln119870max = 2 sdot ln119870 minus ln119870min
According to the area superposition principle of normaldistribution curve and giving a initial value 120590 of normaldistribution then119883
0= ln119870min and119883119899 = ln119870max 119865(119883119895) is a
cumulative distribution function of the normal distribution[13] If we use equal step size Δ119883 make a subdivision for
Journal of Applied Mathematics 5
probability curve then the area summation of every closedfigures 119878
119895= 119865(119883
119895) minus119865(119883
119895minus1) each longitudinal layer stratum
thickness ℎ119895= ℎ sdot 119878
119895 and 119883
119895= Δ119883 sdot 119895 + 119883
0 119870119895= 119890119883119895
if we use equal area Δ119878 make a subdivision for probabilitycurve then the area summation of every closed figures 119878
119895=
119865(119883119895)minus119865(119883
119895minus1) 119878119895= Δ119878 = 1119899 ℎ
119895= ℎsdot119878
119895 and119883
119895= 119865minus1(119878119895)
119870119895= 119890119883119895 And thus 119895 = 1 2 3 119899 (119899 is the total number of
vertical stratification)Relying on the distribution 119870
1 1198702 119870
119899 oil-water
relative permeability functions 119896119903119900 = 119896119903119900(119878119908) 119896119903119908 =
119896119903119908(119878119908) [14] the conductivity definition of numerical reser-
voir simulation we get the following equations
119896119903119900119894
119895= 119896119903119900 (119878
119908
119894(119905)
119895) 119896119903119908
119894
119895= 119896119903119908 (119878
119908
119894(119905)
119895)
119879119894
119895=
2120587ℎ119895119870119895
ln (119903119903119894119895)(119870119903119900
119894
119895
120583119900
+119870119903119908
119894
119895
120583119908
)
119879119895=
1
sum119898
119894=1(1119879119894119895) 119879 =
119899
sum119895=1
119879119895
119876119905
119895=119879119895
119879sdot 119876119905
(17)
We can derive from (12) and (17) that 119903119908119891119895
If 119903119890
le 119903119908119891119895
solving inverse function of (11) andcalculating water saturation of the fluid producing edge then119878119908
119894(119905)
119895= 119865minus1(119903119894
119895 120601119895 ℎ119895 119876119905
119895) and thus 119894 = 1 If 119903
119890gt 119903119908119891119895
then
119878119908
119894(119905)
119895= 119878119908119888119895
and thus 119894 = 1If we obtain the value of the water saturation (119878
119908) relying
on the function119891119908(119878119908) sim 119878119908 we can calculate the value of the
longitudinal each layer water cut 119891119905119908(1198781
119908119895) then mathematic
model can be constructed as follows
119876119905
119908119895= 119876119905
119895sdot 119891119905
119908(1198781
119908119895) 119876
119905
119900119895= 119876119905
119895sdot (1 minus 119891
119905
119908(1198781
119908119895))
119876119905
119908=
119899
sum119895=1
119876119905
119908119895 119876
119905
119900=
119899
sum119895=1
119876119905
119900119895
119891119908(119905) =
119876119905
119908
(119876119905119908+ 119876119905119900)
(18)
Equation (18) can calculate the total water cut of the fluidproducing edge 119891
119908(119905) and relies on the objective function
119864 = minsum119899119905119905=1
(119891119908(119905)minus119891
(history)119908
(119905))2With numerical optimiza-
tion calculation in the inverse problem model the standarddeviation 120590 and an optimization problem of the distributionLn1198701 Ln1198702 Ln119870
119899 will be obtained so the distribution
1198701 1198702 119870
119899 will be given The proof of Theorem 4 is
completed
5 Discussions and Conclusions
The different distribution 119886119908 119887119908 119886119900 119887119900 corresponds to a
group of oil-water relative permeability equations 119870119903119908(119878119908)
and 119870119903119900(119878119908) Combining with the objective function
119864 = minsum119899119905119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2 through the optimization
solution method it will finally bring a set of the optimumdistribution 119886
119908 119887119908 119886119900 119887119900 and the oil-water relative
permeability equations119870119903119908(119878119908) and119870
119903119900(119878119908)
According to the above inverse problem mathematicalmodel based on the definition of the normal distributiondifferent (120583
119894 1205902
119894) corresponds to its own normal distribution
curve with the certain expectation 120583119894 If 120590
119894goes up the
volatility of its normal distribution will be stronger accordingto the definition of standard deviation which will finally leadto the large differential permeability distribution namelythe strong heterogeneity With the certain expectation 120583
119894
each different value of 120590119894corresponds to a group of original
values of (120583119894 1205902
119894) which will yield a group of values of
1198701 1198702 119870
119899minus1 119870119899 namely the value of permeability of
each single formation And it will also work out liquidproduction water production and integrated water cut of thewhole liquid outlet of each single formation Combining withthe objective function 119864 = minsum119899119905
119905=1(119891119908(119905) minus 119891
(history)119908
(119905))2
it will finally bring a set of optimum normal distributionsof 119883 sim (120583
119894 1205902
119894) (Etc 119883 = ln119870) through the optimization
solution method Analysis of the changes of water drivingplace oil production and water production can be donethrough the related mathematical model
Finally the idea of constructing inverse problem modelsaccording to the historical dynamic production data canbe realized which attaches great importance to formationheterogeneity observation of water flooding front positionand prediction of dynamic producing performance
Acknowledgments
The authors gratefully thank the support of Canada CMGFoundation and the referees for valuable suggestions
References
[1] S X Bing ldquoReservoir numerical simulation study based onformation parametersrsquo time-variabilityrdquo Procedia Engineeringvol 29 pp 2327ndash2331 2012
[2] F A Dumkwu A W Islam and E S Carlson ldquoReview ofwell models and assessment of their impacts on numericalreservoir simulation performancerdquo Journal of Petroleum Scienceand Engineering vol 82-83 pp 174ndash186 2012
[3] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011
[4] S Haj-shafiei S Ghosh and D Rousseau ldquoKinetic stability andrheology of wax-stabilized water-in-oil emulsions at differentwater cutsrdquo Journal of Colloid and Interface Science vol 410 pp11ndash20 2013
[5] S Talatori and T Barth ldquoRate of hydrate formation in crudeoilgaswater emulsions with different water cutsrdquo Journal ofPetroleumScience and Engineering vol 80 no 1 pp 32ndash40 2011
[6] S Ji C Tian C Shi J Ye Z Zhang and X Fu ldquoNewunderstanding on water-oil displacement efficiency in a highwater-cut stagerdquo Petroleum Exploration and Development vol39 no 3 pp 362ndash370 2012
6 Journal of Applied Mathematics
[7] F Doster and R Hilfer ldquoGeneralized Buckley-Leverett theoryfor two-phase flow in porous mediardquo New Journal of Physicsvol 13 Article ID 123030 2011
[8] J Wang P Yang and Q Liu ldquoDetermination of relativepermeability curves using data measured with rate-controlledmercury penetrationrdquo Journal of the University of PetroleumChina vol 27 no 4 pp 66ndash69 2003
[9] E DiBenedetto U Gianazza and V Vespri ldquoContinuity of thesaturation in the flow of two immiscible fluids in a porousmediumrdquo Indiana University Mathematics Journal vol 59 no6 pp 2041ndash2076 2010
[10] S S Al-Otaibi and A A Al-Majed ldquoFactors affecting pseudorelative permeability curvesrdquo Journal of Petroleum Science andEngineering vol 21 no 3-4 pp 249ndash261 1998
[11] W Littmann and K Littmann ldquoUnstructured grids for numer-ical reservoir simulationmdashusing TOUGH2 for gas storagerdquo OilGas-European Magazine vol 38 pp 204ndash209 2012
[12] I Brailovsky A Babchin M Frankel and G SivashinskyldquoFingering instability in water-oil displacementrdquo Transport inPorous Media vol 63 no 3 pp 363ndash380 2006
[13] D Bonnery F J Breidt and F Coquet ldquoUniform convergenceof the empirical cumulative distribution function under infor-mative selection from a finite populationrdquo Bernoulli vol 18 no4 pp 1361ndash1385 2012
[14] T Ramstad N Idowu C Nardi and P E Oslashren ldquoRelativepermeability calculations from two-phase flow simulationsdirectly on digital images of porous rocksrdquo Transport in PorousMedia vol 94 no 2 pp 487ndash504 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
probability curve then the area summation of every closedfigures 119878
119895= 119865(119883
119895) minus119865(119883
119895minus1) each longitudinal layer stratum
thickness ℎ119895= ℎ sdot 119878
119895 and 119883
119895= Δ119883 sdot 119895 + 119883
0 119870119895= 119890119883119895
if we use equal area Δ119878 make a subdivision for probabilitycurve then the area summation of every closed figures 119878
119895=
119865(119883119895)minus119865(119883
119895minus1) 119878119895= Δ119878 = 1119899 ℎ
119895= ℎsdot119878
119895 and119883
119895= 119865minus1(119878119895)
119870119895= 119890119883119895 And thus 119895 = 1 2 3 119899 (119899 is the total number of
vertical stratification)Relying on the distribution 119870
1 1198702 119870
119899 oil-water
relative permeability functions 119896119903119900 = 119896119903119900(119878119908) 119896119903119908 =
119896119903119908(119878119908) [14] the conductivity definition of numerical reser-
voir simulation we get the following equations
119896119903119900119894
119895= 119896119903119900 (119878
119908
119894(119905)
119895) 119896119903119908
119894
119895= 119896119903119908 (119878
119908
119894(119905)
119895)
119879119894
119895=
2120587ℎ119895119870119895
ln (119903119903119894119895)(119870119903119900
119894
119895
120583119900
+119870119903119908
119894
119895
120583119908
)
119879119895=
1
sum119898
119894=1(1119879119894119895) 119879 =
119899
sum119895=1
119879119895
119876119905
119895=119879119895
119879sdot 119876119905
(17)
We can derive from (12) and (17) that 119903119908119891119895
If 119903119890
le 119903119908119891119895
solving inverse function of (11) andcalculating water saturation of the fluid producing edge then119878119908
119894(119905)
119895= 119865minus1(119903119894
119895 120601119895 ℎ119895 119876119905
119895) and thus 119894 = 1 If 119903
119890gt 119903119908119891119895
then
119878119908
119894(119905)
119895= 119878119908119888119895
and thus 119894 = 1If we obtain the value of the water saturation (119878
119908) relying
on the function119891119908(119878119908) sim 119878119908 we can calculate the value of the
longitudinal each layer water cut 119891119905119908(1198781
119908119895) then mathematic
model can be constructed as follows
119876119905
119908119895= 119876119905
119895sdot 119891119905
119908(1198781
119908119895) 119876
119905
119900119895= 119876119905
119895sdot (1 minus 119891
119905
119908(1198781
119908119895))
119876119905
119908=
119899
sum119895=1
119876119905
119908119895 119876
119905
119900=
119899
sum119895=1
119876119905
119900119895
119891119908(119905) =
119876119905
119908
(119876119905119908+ 119876119905119900)
(18)
Equation (18) can calculate the total water cut of the fluidproducing edge 119891
119908(119905) and relies on the objective function
119864 = minsum119899119905119905=1
(119891119908(119905)minus119891
(history)119908
(119905))2With numerical optimiza-
tion calculation in the inverse problem model the standarddeviation 120590 and an optimization problem of the distributionLn1198701 Ln1198702 Ln119870
119899 will be obtained so the distribution
1198701 1198702 119870
119899 will be given The proof of Theorem 4 is
completed
5 Discussions and Conclusions
The different distribution 119886119908 119887119908 119886119900 119887119900 corresponds to a
group of oil-water relative permeability equations 119870119903119908(119878119908)
and 119870119903119900(119878119908) Combining with the objective function
119864 = minsum119899119905119905=1
(119891119908(119905) minus 119891
(history)119908
(119905))2 through the optimization
solution method it will finally bring a set of the optimumdistribution 119886
119908 119887119908 119886119900 119887119900 and the oil-water relative
permeability equations119870119903119908(119878119908) and119870
119903119900(119878119908)
According to the above inverse problem mathematicalmodel based on the definition of the normal distributiondifferent (120583
119894 1205902
119894) corresponds to its own normal distribution
curve with the certain expectation 120583119894 If 120590
119894goes up the
volatility of its normal distribution will be stronger accordingto the definition of standard deviation which will finally leadto the large differential permeability distribution namelythe strong heterogeneity With the certain expectation 120583
119894
each different value of 120590119894corresponds to a group of original
values of (120583119894 1205902
119894) which will yield a group of values of
1198701 1198702 119870
119899minus1 119870119899 namely the value of permeability of
each single formation And it will also work out liquidproduction water production and integrated water cut of thewhole liquid outlet of each single formation Combining withthe objective function 119864 = minsum119899119905
119905=1(119891119908(119905) minus 119891
(history)119908
(119905))2
it will finally bring a set of optimum normal distributionsof 119883 sim (120583
119894 1205902
119894) (Etc 119883 = ln119870) through the optimization
solution method Analysis of the changes of water drivingplace oil production and water production can be donethrough the related mathematical model
Finally the idea of constructing inverse problem modelsaccording to the historical dynamic production data canbe realized which attaches great importance to formationheterogeneity observation of water flooding front positionand prediction of dynamic producing performance
Acknowledgments
The authors gratefully thank the support of Canada CMGFoundation and the referees for valuable suggestions
References
[1] S X Bing ldquoReservoir numerical simulation study based onformation parametersrsquo time-variabilityrdquo Procedia Engineeringvol 29 pp 2327ndash2331 2012
[2] F A Dumkwu A W Islam and E S Carlson ldquoReview ofwell models and assessment of their impacts on numericalreservoir simulation performancerdquo Journal of Petroleum Scienceand Engineering vol 82-83 pp 174ndash186 2012
[3] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011
[4] S Haj-shafiei S Ghosh and D Rousseau ldquoKinetic stability andrheology of wax-stabilized water-in-oil emulsions at differentwater cutsrdquo Journal of Colloid and Interface Science vol 410 pp11ndash20 2013
[5] S Talatori and T Barth ldquoRate of hydrate formation in crudeoilgaswater emulsions with different water cutsrdquo Journal ofPetroleumScience and Engineering vol 80 no 1 pp 32ndash40 2011
[6] S Ji C Tian C Shi J Ye Z Zhang and X Fu ldquoNewunderstanding on water-oil displacement efficiency in a highwater-cut stagerdquo Petroleum Exploration and Development vol39 no 3 pp 362ndash370 2012
6 Journal of Applied Mathematics
[7] F Doster and R Hilfer ldquoGeneralized Buckley-Leverett theoryfor two-phase flow in porous mediardquo New Journal of Physicsvol 13 Article ID 123030 2011
[8] J Wang P Yang and Q Liu ldquoDetermination of relativepermeability curves using data measured with rate-controlledmercury penetrationrdquo Journal of the University of PetroleumChina vol 27 no 4 pp 66ndash69 2003
[9] E DiBenedetto U Gianazza and V Vespri ldquoContinuity of thesaturation in the flow of two immiscible fluids in a porousmediumrdquo Indiana University Mathematics Journal vol 59 no6 pp 2041ndash2076 2010
[10] S S Al-Otaibi and A A Al-Majed ldquoFactors affecting pseudorelative permeability curvesrdquo Journal of Petroleum Science andEngineering vol 21 no 3-4 pp 249ndash261 1998
[11] W Littmann and K Littmann ldquoUnstructured grids for numer-ical reservoir simulationmdashusing TOUGH2 for gas storagerdquo OilGas-European Magazine vol 38 pp 204ndash209 2012
[12] I Brailovsky A Babchin M Frankel and G SivashinskyldquoFingering instability in water-oil displacementrdquo Transport inPorous Media vol 63 no 3 pp 363ndash380 2006
[13] D Bonnery F J Breidt and F Coquet ldquoUniform convergenceof the empirical cumulative distribution function under infor-mative selection from a finite populationrdquo Bernoulli vol 18 no4 pp 1361ndash1385 2012
[14] T Ramstad N Idowu C Nardi and P E Oslashren ldquoRelativepermeability calculations from two-phase flow simulationsdirectly on digital images of porous rocksrdquo Transport in PorousMedia vol 94 no 2 pp 487ndash504 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
[7] F Doster and R Hilfer ldquoGeneralized Buckley-Leverett theoryfor two-phase flow in porous mediardquo New Journal of Physicsvol 13 Article ID 123030 2011
[8] J Wang P Yang and Q Liu ldquoDetermination of relativepermeability curves using data measured with rate-controlledmercury penetrationrdquo Journal of the University of PetroleumChina vol 27 no 4 pp 66ndash69 2003
[9] E DiBenedetto U Gianazza and V Vespri ldquoContinuity of thesaturation in the flow of two immiscible fluids in a porousmediumrdquo Indiana University Mathematics Journal vol 59 no6 pp 2041ndash2076 2010
[10] S S Al-Otaibi and A A Al-Majed ldquoFactors affecting pseudorelative permeability curvesrdquo Journal of Petroleum Science andEngineering vol 21 no 3-4 pp 249ndash261 1998
[11] W Littmann and K Littmann ldquoUnstructured grids for numer-ical reservoir simulationmdashusing TOUGH2 for gas storagerdquo OilGas-European Magazine vol 38 pp 204ndash209 2012
[12] I Brailovsky A Babchin M Frankel and G SivashinskyldquoFingering instability in water-oil displacementrdquo Transport inPorous Media vol 63 no 3 pp 363ndash380 2006
[13] D Bonnery F J Breidt and F Coquet ldquoUniform convergenceof the empirical cumulative distribution function under infor-mative selection from a finite populationrdquo Bernoulli vol 18 no4 pp 1361ndash1385 2012
[14] T Ramstad N Idowu C Nardi and P E Oslashren ldquoRelativepermeability calculations from two-phase flow simulationsdirectly on digital images of porous rocksrdquo Transport in PorousMedia vol 94 no 2 pp 487ndash504 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of